Problem 67 - Physics for Scientists and Engineers with Modern Physics

Question

A coil with 150 turns, a radius of \(5.0 \mathrm{~cm},\) and a resistance of \(12 \Omega\) surrounds a solenoid with 230 turns \(/ \mathrm{cm}\) and a radius of \(4.5 \mathrm{~cm}\); see Fig. \(29-50 .\) The current in the solenoid changes at a constant rate from 0 to \(2.0 \mathrm{~A}\) in \(0.10 \mathrm{~s}\). Calculate the magnitude and direction of the induced current in the 150 -turn coil.

Step-by-Step Solution

Answer

The induced current has a magnitude of approximately 80.8 mA and flows counterclockwise.

Step 1: Determine Change in Magnetic Flux

First, compute the change in the magnetic flux through the coil. Since the solenoid has 230 turns per cm, this means it has 2300 turns per meter. The magnetic field inside the solenoid can be expressed as \( B = \mu_0 n I \), where \( n = 23000 / \text{m} \), and \( I \) is the current. Changing the current from 0 to 2 A in 0.1 s, the change in magnetic flux \( \Delta \Phi \) is \( \Delta B \cdot A \) where \( \Delta B = \mu_0 n \Delta I \) and \( A = \pi r^2 \) is the area of the solenoid. Thus, we have:\[ \Delta \Phi = (\mu_0 \cdot 23000 \cdot \pi \cdot (0.045)^2 \cdot 2.0) \]

Step 2: Calculate Induced Electromotive Force (EMF)

Using Faraday's law of induction, the magnitude of the induced EMF is given by \( \text{EMF} = -N \frac{\Delta \Phi}{\Delta t} \), where \( N = 150 \) is the number of turns in the coil, and \( \Delta t = 0.10 \hspace{1mm} s \) is the time over which the current changes. Plug in the value for \( \Delta \Phi \) from the previous step:\[ \text{EMF} = -150 \cdot \frac{\mu_0 \cdot 23000 \cdot \pi \cdot (0.045)^2 \cdot 2.0}{0.10} \]

Step 3: Calculate Induced Current

The induced current can be found using Ohm's Law \( I = \frac{\text{EMF}}{R} \), where \( R = 12 \hspace{1mm} \Omega \) is the resistance of the coil. Substitute the expression for \( \text{EMF} \):\[ I = \frac{-150 \cdot \mu_0 \cdot 23000 \cdot \pi \cdot (0.045)^2 \cdot 2.0}{0.10 \cdot 12} \]

Step 4: Calculate the Magnitude and Direction

Compute the magnitude using the expression from the previous step. The direction of the induced current is given by Lenz's Law, which indicates that it will act to oppose the change that produced it. Since the current in the solenoid increased, the induced current in the coil will flow in a direction to create a magnetic field opposing the solenoid's increase, hence it will be counterclockwise when seen from the solenoid's side.

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle describing how electric current is generated in a coil by changing the magnetic field around it. Think of it as nature's way of converting magnetic energy into electrical energy. According to Faraday's Law, the induced electromotive force (EMF) in a coil is directly proportional to the rate of change of magnetic flux through the coil. This experiment can be seen in our exercise where a solenoid induces an EMF in a nearby coil.

The formula for Faraday's Law is expressed as:

• \[ \text{EMF} = -N \frac{\Delta \Phi}{\Delta t} \]

Where:

• \( N \) is the number of turns in the coil.
• \( \Delta \Phi \) represents the change in magnetic flux.
• \( \Delta t \) is the time interval over which the change occurs.

In the problem, this law helps us calculate the EMF by considering the change in magnetic flux, which is crucial in determining the induced current. The negative sign in the formula signifies Lenz's Law, which we will cover next.

Lenz's Law

Lenz's Law provides essential information about the direction of the induced current. It is crucial in understanding the nature of electromagnetic induction since it states that the induced current will always work to oppose the change in magnetic flux that caused it.

This concept can be experienced in cases such as our exercise, where the current in the solenoid changes. Because the solenoid's magnetic field increases, Lenz's Law predicts that the induced current in the surrounding coil will create a magnetic field opposing this increase. In this scenario, the induced current flows in the counterclockwise direction if you view it from the direction of the solenoid. Thanks to Lenz's Law, we can always predict the correct direction of the induced current based on the original cause of magnetic flux change.

Ohm's Law

Ohm's Law connects the relationship between voltage, current, and resistance in electric circuits, and it is one of the foundational principles of electronics. The law explains how potential difference is necessary to drive an electric current through a resistor or any conducting material. Formally, Ohm's Law states:

• \[ I = \frac{\text{EMF}}{R} \]

Where:

• \( I \) is the induced current.
• \( \text{EMF} \) is the electromotive force induced in the coil.
• \( R \) is the resistance in the coil.

In the problem, we apply Ohm's Law to calculate the magnitude of the induced current after determining the EMF using Faraday's Law. By knowing the coil's resistance, you can easily find the induced current, making it a practical piece of the puzzle for solving real-world problems in electromagnetic induction.

Magnetic Flux

Magnetic flux measures how much magnetic field passes through a specified area. You can imagine it as counting the number of magnetic field lines that pass through a surface, like thread through a sewing needle.

In mathematical terms, the magnetic flux \( \Phi \) through an area \( A \) is defined as:

• \[ \Phi = B \times A \times \cos(\theta) \]

Where:

• \( B \) is the magnetic field strength.
• \( A \) is the perpendicular area through which the lines pass.
• \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface.

In our exercise, the coil experiences a change in \( \Phi \) because of the solenoid's varying current, and it is this change that induces EMF according to Faraday's Law. Understanding magnetic flux allows us to connect the physical alteration of magnetic conditions with the electrical outcomes observed in circuits.